CA Geometry: Deducing Angle Measures 46-50, deducing the measure of angles
CA Geometry: Deducing Angle Measures
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- We're on problem 46.
- In the figure below, line segment AB is parallel to C.
- I think there's a typo here.
- I've got to believe this has got to be CD.
- That's got to be a typo.
- All right, so AB, I'm assuming, is parallel to CD.
- That's probably what they wanted to have, because you
- can't be parallel to a vertex.
- And they say, what is the value of x?
- So the way to think about it is, if these two are parallel
- lines, then this segment up here is a transversal.
- If we just extend the two lines like this, let me see if
- I can do it well.
- So if I extended that line like that, and if I extended
- this line like that, then segment AD is just a
- Let me see if I can do that one right.
- So if I just extended it like that, you see it's a
- transversal, and I could go the other direction as well.
- I think you get the idea.
- So if that is a transversal, what do you know about
- We know that this angle right here, I'll draw it small, is
- congruent to this angle over here.
- So the measure of this angle is also x plus 40.
- Because they're corresponding angles, and you could see that
- by inspection, and if you moved around the transversal,
- it would make sense that that's the case.
- So this is x plus 40 and this is minus 40, and they're
- clearly supplements of each other.
- They're supplementary angles.
- Then the sum of these two angles have
- to be equal to 180.
- So let's figure it out.
- So x minus 40 plus x plus 40 is equal to 180, because
- they're supplementary.
- The 40's cancel out.
- So you just minus 40, plus 40 adds up to zero.
- So you're left with 2x is equal to 180.
- x is equal to 90.
- So it's D.
- 47: The measures of the interior angles of a pentagon
- are 2x, 6x, 4x minus 6, 2x minus 16, and 6x plus 2.
- What is the measure in degrees of the largest angle?
- OK, so first of all, we have to remember what is the sum of
- the interior angles of a pentagon?
- And that's where I always draw an arbitrary pentagon.
- Let me see if I can do that.
- Actually, there's a polygon tool here.
- How does it work?
- I'm just trying to draw a pentagon.
- I don't know if that's any different than the line tool,
- but regardless.
- So how many triangles can I draw in a pentagon?
- And that tells me what my total interior angles are.
- And there is a formula for that, but I like relying on
- your reasoning more than the formula, because you might
- forget the formula, or even worse, you might remember it,
- but not have the confidence to use it, or you might remember
- it wrong ten years in the future.
- So the best thing to do, if you have a polygon, is to
- count the triangles in it.
- Straightforward enough.
- It's almost easier than using the formula.
- So a pentagon has three triangles in it.
- So the sum of its interior angles are going to be 3 times
- 180, because it has 3 triangles in it.
- Each triangle has 180 degrees.
- And I know you can't see what I just wrote.
- So the sum of all of these angles are going to be the sum
- of all of the angles in all three triangles.
- So it's 3 times 180 is equal to 540 degrees.
- So that's the sum of all of these interior angles.
- Now, they say that each of them are 2x, 6x, et cetera.
- So the sum of all of these terms have to be equal 540.
- I'm going to write them vertically.
- It makes them easier to add.
- So if we write there 2x, 6x, 4x minus 6, 2x minus
- 16, and 6x plus 2.
- This is going to be the largest, right?
- That sum is going to equal 540.
- So let's add this up.
- Minus 6, minus 16, that's minus 22.
- Plus 2 is minus 20.
- That's right.
- And 2x plus 6x that is 8x plus 4 is 12, 12 plus 2 is 14, 14
- plus 6 is 20.
- So we have 20x minus 20 is equal to 540 degrees.
- Let me write it again.
- 20x minus 20 is equal to 540.
- Let's divide both sides of this equation by 20.
- So you get x 1 minus 1 is equal to-- it would be 54
- divided by 2, which is equal to 27.
- Add 1 to both sides, x is equal to 28.
- And they want to know, what is a measure in degrees of the
- largest angle?
- That's going to be this one.
- That's the largest one.
- It's 6 times x plus 2.
- So 6 times 28, that's 48.
- 2 times 6 is 12 plus 4 is 168.
- So it's 168 plus 2.
- It's 170 degrees.
- Choice C.
- Problem 48: What is the measure of angle 1?
- So this we're going into the angle game.
- And these are fun, because they are kind of these
- deductive reasoning problems where you just use a couple of
- simple rules and just fill in the whole thing.
- So let's think about it.
- This is 36 degrees.
- They tell us that this whole angle right
- here is a right angle.
- So this angle right here is going to be the complement to
- 36 degrees.
- 36 plus this angle have to be equal to 90.
- So what's this one?
- This one is 90 minus 36, which is 54.
- That's going to be 54 degrees.
- 90 minus 30 is 60.
- Right, that's 54.
- And this angle right here, that's going to be the
- supplement of 88.
- So this is going to be-- I'll do it in a different color--
- 180 minus 88.
- That is equal to 92 degrees.
- Now this angle 1 plus the 54 plus the 92 is equal to 180.
- So we know that-- let's say angle one plus 54 plus 92 is
- equal to 180.
- This is 146 is equal to 180.
- Subtract 146 from both sides.
- The measure of angle one is equal to 80 minus 40 is 40, so
- 80 minus 46 is equal to 34 degrees.
- So the answer is A.
- Problem 49: What is the measure of angle WZX?
- So they want to know what this angle right here is.
- Let's do the angle game some more.
- Let's see, we can immediately figure out what this angle is,
- because it is the supplement of 132 degrees, so this is
- going to be 180 minus 132.
- So this is 48 degrees.
- This angle plus this angle is going to be
- equal to this angle.
- Or this angle plus this angle plus this
- angle is equal to 180.
- I don't know what I just said, I think I
- said something wrong.
- Write that down.
- So this angle is going to be equal to 180
- minus 52 minus 48.
- Because the sum of the angles add up to 180, and so that is
- equal to 180 minus 100 which equals 80 degrees.
- So this angle right here is equal to 80 degrees.
- And the angle they want us to figure out is the opposite of
- this angle, or in the U.S., I guess, they
- say vertical angles.
- And so opposite or vertical angles are equal or they're
- congruent, so this is going to be 80 degrees as well.
- And that is choice A.
- Problem 50: What is the measure of an exterior angle
- of a regular hexagon?
- A regular hexagon tells us that all of the sides are the
- same, it's equilateral, and all of the angles are the
- same, equiangular.
- So if we just knew what's the total degree measure of the
- interior angles, we could just divide that by 6, and then
- that would give us what each of the interior angles are,
- and then we could use that information to figure out the
- exterior angles.
- Let's just do it.
- So once again, I like to just draw a hexagon.
- Let's just draw a hexagon and count the triangles in it.
- Two sides, three sides, four sides, five
- sides and six sides.
- And how many triangles do I have here?
- One, two, three.
- So I have one, two, three, four triangles.
- The sum of the interior angles of this hexagon, of any
- hexagon, whether it's regular or not, are going to be 4
- times 180 and that's 720 degrees.
- And it's a regular hexagon, so all the interior angles are
- going to be the same.
- And there's six of them.
- So each of them are going to be 720 divided by 6.
- Well, 6 goes into 72 twelve times.
- So each of the interior angles are going to be 120 degrees.
- And I didn't draw it that regular, but we can assume
- that all of these are each 120 degrees.
- Fair enough.
- Now, if all of those are each 120 degrees, what is the
- measure of an exterior angle?
- Well, we could just extend one of these sides
- out a little bit.
- We could say, OK, if this is 120 degrees, what is its
- Well, these have to add up to 180, so 180
- minus 120 is 60 degrees.
- I could do it on any side.
- I could extend that line out there, and I'd say, oh, that's
- 60 degrees.
- So any of the exterior angles are 60 degrees.
- All right, do I have time for one more?
- I'll wait for the next one in the next video.
- See you soon.
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